1) Is there a difference between justice and revenge?
Yes. Justice is about being just. Revenge is about making someone pay for what they did, in many cases it involves extracting a bigger cost. It is about personal gratification and dominance.
I think revenge and justice get mixed because both are related to fairness. The reason a person feels retaliating is because they don't feel justified the original blow. Since justice is about fairness many people go black and white ona situation and expects an eye for an eye kind of solution to be synonymous with justice.
This reminds me of the King Solomon and the tale taht scared me off as a kid.
"King Solomon"
I might be a bit off in details but the gist is the following: Two women, one with a baby of a few days old, entered the court of the King to solve their problem. They were living under the same roof they had given birth around the same time but one of the babies had died the night before. The one without the creature claimed to be the real mother and that the other woman had stolen her baby and exchanged for the corpse. The woman with the kid kept assuring to be the real mother and that the first woman was trying to keep her baby because she lost her own. So King Solomon called for a sword and gave the verdict to divide the baby and give a half to each. Soon enough once of the woman cried out for the baby's life and asked to just give the baby to the other woman. Solomon then decided she was the real mother and gave the son to her.
Even though the bible version makes clear she was actually the real mother and the other woman still bitter with her own baby death as being fine with the split verdict, I don't think it is what really matters. We have two people claiming to be the parent and one baby. The way justice is imparted here is that without doubt the baby ended up in the arms of the person that cared for the creature the most. It was never the main point to divide the child but to create a gut response that revealed what kind of feelings they had for the baby. The kid would have a better future that way.
I think that in general revenge is a negative mindset because it means you're not moving from situations and letting your pride being your pilot. The bitterness that comes with it is a natural human emotion that would be unhealthy to bottle and ignore. Revenge fantasies are a normal coping mechanism. So in itself the idea of revenge seems to be a part of humanity to me. The intention and severity of the actions moved by revenge are the issue and not the revenge in itself. If said revenge means the harm of one or more individuals then that form of revenge is not okay.
So for example, if killing is wrong, killing for revenge is still wrong.
I didn't really connect Justice with punishment until the question. I don't think we can answer a contundent yes or a contundent no. It depends on the situation, the punishment and what it is the point of it. Negative reinforcement is one of the reasons you get a punishment and in the basic level it works. You associate the bad action with pain. The problem with this comes that it is not really changing someone's mind on something as mucha s training them to control their impulse toward certain things, or at least is the intention. Punishing a criminal is meant to have them stride away from crime but instead for many the lesson becomes "don't get caught". It does nothing for a person to understand why it was wrong or come to a inner resolution to stop it.
Killing a killer. Is it revenge or justice? Debate has been going on for a long time and I have no clear stance on it being the right or the wrong thing to do. Do you kill the murder to avoid more innocent bloodshed? Do you kill them because it is a life for a life penalty?Do you kill them to show dominance over other people that might be thinking on killing?
Should someone who killed but shows remorse been treated any better than that of an unrepentant one?
I don't know. I just feel taht when I was a kid Justice was a crystal clear black and white situation but as I grew up became this complex things where my beliefs became challenged by books, video games and other media.